From: AIPS 2000 Proceedings. Copyright © 2000, AAAI (www.aaai.org). All rights reserved.
PSIPLAN: Open World Planning
Tamara Babaian
Dept.
ofElectrical
Eng.and
Computer
Science
~ifts
University
Medford.MA 02155USA
tbabalan,~eecs.tufts.edu
Abstract
Wepresent a new,partial order planner called PSIPLAN,which builds on SNLP.Wedrop the closed
worldassumption,addsensingactions, "~ld a class of
propositions about the agent’s knowledge,and add a
class of universallyquantifiedpropositions.Thislatter class of propositions,whichwecall C-forms,distinguishesthis research. ~,-formsrepresentpartially
closedworlds,suchas "’Block.4 is clear", or "z.ps is
the only postscript file in directory/tez."Wepresent
our theory"of planningwith sensingandshow,hob-partial orderplanningis performed
usingI/~-forms.Noteworthyare the facts that lack of information
canbe
representedpreciselyandall quantifiedreasoninghas
polynomialcomplexity.Thus,in finite domainswhere
the maximum
plan length is bounded,planning with
PSIPLAN
is NP-complete.
Introduction
Several researchers have examinedplanning with sensing in an open world, where the agent does not have
complete information about the world and must take
~tions both to acquire knowledgeand to change the
world (e.g., (Peot & Smith 1992), (Etzioni et al.
1992), (Krebsbach, Olawsky,& Gin, 1992),(Scherl
Levesque1997),(Golden1998)). Incompletenessof
agent’s knowledgemeansthat it cannot use the Closed
WorldAssumption(CWA)in the representation of the
state and must rely on a different methodfor representing large quantities of negative information. The
most comprehensiveof all practical solutions to this
problem is the PUCCINIplanner (Golden 1998).
uses the LCWsentences in its representation, which
weanalyze in the next section.
Someworks(e.g.,
(Scherl
& Levesque
1997))
first-order
logic
(FOL),
which
easily
represents
open
worlds,
butwhich
appears
topreclude
practical
planningalgorithms
duetotheundecidability
ofentalhnent
in FOL.Conformant
Graphplan
(Smith
& Weld1998)
considers
eachpossible
world.
SGP(Weld& Anderson1998)extends
Conformant
Graphplan
to handling
with
-forms.
James G. Schmolze
Dept. of Electrical Eng. and
Computer
Science
Tufts
University
Medford,MA 02155USA
schmolze~eecs.tufts.edu
http://www.eccs.tufts.edu/~schmolze
sensing actions and uncertain effects.
Wehave developed a planning formalismcalled PSIPLAN
and a sound and complete partial order plmmer
(POP) called PSIPOPthat uses PSIPLAN
to plan
open worlds without sensing. Moreover, we have extended both PSIPLANand PSIPOPto handle sensing
actions, knowledgegoals, information loss and conditional effects.
In this paper, we focus on demonstratingthe power
of our ~-form-based language PSIPLAN,discussing
the issues critical to its soundnessand completeness
in open world planning, and extending the standard
POPalgorithm to produce PSIPOP.
Representing Open Worlds
We consider
theproblem
ofopenworld
planning
where
theagentdoesnothavecomplete
information
about
theworld.
We assume
thattheworldevolves
as a
sequence
ofstates,
wherethetransitions
occur
only
astheresult
ofdeliberate
action
taken
bythesingle
agent.
Since the agent’s modelof the world is incomplete,
wemust distinguish betweenthe world state (or state
of the world, or situation, in situation calculus terms
(McCarthy& Hayes1969)) and the state of the agent’s
knowledgeof the world, which we call SOK.Wefurther assumethat the agent’s knowledgeof the world is
correct.
While in theory the numberof propositions in an
SOKcan be unlimited, for practical purposes it must
be finite and preferably as small as possible. In many
domainsthe numberof negated propositions that are
true in a worldstate is usually very large, if not infinite. Wecannot use the CWA,because we must distinguish propositions that are false fromthose that are
unknown.To compactlyreprescnt such negated informationone solution is to use quantified formulas~
t~,Ve use the term "quantified formula"informallyto
refer to anyformulathat can representa possiblyinfinite
C,,lvright 0 2000. AmericanAssociation Ibr
Aciificial Intelligence. All rigl’us reserved.
292 AIPS-2000
From: AIPS 2000 Proceedings. Copyright © 2000, AAAI (www.aaai.org). All rights reserved.
(Etzioni, Golden, &Weld1997) define a special class
of LCW(for Local Closed World information)
sentences, to specify the parts of SOKfor which information is complete. The agent is said to have Local
Closed World information relative to a logical formula
if the value of every ground sentence that unifies with
is either knownto be true or knownto be false. For
example, LCW(PS(x) In (x,/rex))
st ates th at th e
agent knowsall x’s that are postscript files in directory ~rex, i.e. given any particular x, the agent knows
whether (PS(x) A In(z, ~rex)) is true or false, but is
not unknown.
There are drawbacks in the LCWrepresentation.
LCWreasoning is incomplete and, in some cases, discards information due to its inability to represent exceptions, i.e. the inability to state that the agent knows
the value of all instantiations of formula ¯ except some.
As the result, when a value of even one instance of
becomes unknown, the entire LCWsentence is no
longer true and the agent must discard it, losing information about those instances that were known to be
false.
In contrast to LCW,we define a class of formulas,
called ~#-forms, that can represent Locally Closed
World Information with exceptions, or what we call
Partially
Closed Worlds. We begin with an example before formally defining ~#-forms.
Example 1 Consider a #)-form
¢ =[ PS(z)v -In(z,y) I-,(x = a.ps)= fig) ].
¢ represents
all clauses of the form -~PS(x)V
-,In(z,/tex) for all values of z except for -,PS(a.ps)
-,In(a.ps, /rex) and -~PS(fig) V-,In(fig, [tex). Given
the following SOK
f
PS(a.ps), In(a.ps,/rex), In(fig,/rex),
s = [-,PS(x) V -,In(x,/tex)]
-,(x = a.ps) A -,(x = fig)].
the agent can conclude that it knows that there are
no postscript files in/tex except for the postscript file
a.ps, and maybethe file fig, whose format is unknown.
Note, we do not have LCW(PS(x) A In(z,/rex))
in
s, because the format of fig is unknown.This example
cannot be represented in LCWrepresentation unless
the domain is finite and ground facts are represented
explicitly, with no LCWsentences.
xb-forms are formulas that are used to represent possibly infinite sets of ground clauses that are obtained
by instantiating what we call its main par~ in all possible ways, except for certain instantiations, listed as
its exceptions. In Example1, the main part ore is the
formula -,PS(x)v-~ln(x,/rex),
while the exceptions
are -~(x = a.ps) and -,(x = fig). For the efficiency of
set of ground formulas
reasoning we choose the main part of C-forms to be a
disjunction of negated literals.
Wecan alternatively view C-forms as logical propositions by interpreting them as a conjunction of all
ground clauses they represent. Weuse this duality
and define both set-theoretic and logical relations between C-forms. This allows us to use them efficiently
in partial order planning.
In addition to the suitability of C-forms for representing negative information, C-form reasoning has
nice computational properties: it is sound, complete in
what we later define as su~ciently large domains, and
has only polynomial complexity.
Definitions
The general form of a C-form is:
¢ = [q(~
l-,a~
A...A -~m,]
(1)
Here, Q(3) is a clause of negated literals that uses
all and only the variables in 3, i.e., Q(~ = --Ql(X~)
... v "-Qk(x~) where k > 0 and each Qi(x~) is
atom that uses all and only the variables in x~ and
k
-- Ui=l :~" Werequire, that none of Qi(~) and
Qj(x~) unify, assuming of course i ~ j.2 In addition,
we require, that for every atom Qi(x~), every variable
in ~ occur no more than once in its argument list.
Thus, [-,P(z, y) v -,Q(y) l-,(x a)] is a C-f orm, while
[-,P(x,x) V -,Q(y) l-,(x a)]is n ot because of mult iple occurances of x in P(x, x).
Each al represents a set of exceptions and is just
a substitution
~ = ~ for some non-empty vector of
variables ~ C_ 3 and some vector of constants e~. Each
must be the same size as its corresponding ~. k and
n are, of course, finite.
We call a C-form with no exceptions a simple Cform. A simple C-form with no variables is called a
singleton and represents a single clause. A C-form
that uses variables is called quantified. Given a Cform (1), we define the following.
¯ .M(¢) is the main part of ¢, Q(3).
¯ V(¢) denotes the variables of 0, 3, though we usually treat it as a set.
¯ r.(~b) is the formula that describes the exceptions,
"mO’l
A ...
n.
A -,~r
¯ ~i(¢) is the substitution for the i-th exception, ai.
¯ #~(~#) is the number of exceptions,
¯ £~(¢) denotes the instantiation ~4(¢)a~.
For example, let
¢ = [-,P(x, y) V ~Q(y, A) I -,(x = A) A -,(x = C,
2Thisrequirementis not critical, but it simplifies calculation of entailment and other operations.
Babaian
293
From: AIPS 2000 Proceedings. Copyright © 2000, AAAI (www.aaai.org). All rights reserved.
¯ v(o)= {~v}
¯ ~C~)= -Cx= 4) ^ -(x = c v D)
¯ Z1(¢) = (x = A}, E2(.~’) = C,y= D},
¯ #ECq,)
= 2,
¯ Et (t~,) = --,P(A, y) V --,Q(y, A), E2(~b) = ~P(C,
-,Q(D, A).
A 0-form is well-formed if and only if there is
no exception that is a subset of another exception,
i.e. .Ei(0) ~ Zj(0), foralli,j
where0 i,j <
#£(~9) and i #
~b-forms as Sets
A tp-form is a representation of a possibly infilfite set
of ground clauses. A set represented by a 0-form is
called a 1b-set. The 0-set corresponding to the ~p-form
~’ is defined with operation $. Wedefine $ recursively
as follows:
1. Wefirst define ~b for a single ~b-form.
Ca) ~([ ]) =
gro und}, if
~b is sim(b) $(~b) I, Vth0)o is
ple. Note that $([c]) = {c}, if c is a groundclause.
Co) ~(~) = ~([..~(~)])
- ,([£1(~)])
~b([E#e(,) (~)]), otherwise.
2. ¢({~’t,... ,~Pk}) = ULt¢(qJi), i.e. a S-set of a set
~9-formsis the union of the ~-sets of its elements.
3. Let [31 and [32 denote either a single ~b-formor a set
of t b-forms. Let * denote any of the set operations
N, U, -, t, or - (last two operations are defined later)
$(F’I 1 * 02) = ~(F’ll) * ~b(F’]2).
Forthesimplicity
ofpresentation,
since~b-forms
and
setsof~b-forms
represent
setsof ground
clauses,
wesay
thata ground
clause
c is in[],written
c E [],instead
of sayingc E $([]).Here[] denotes
eithera single
~b-form
ora setofIp-forms.
Thus,given1.and2.
¯ c E ~’ iff 3~.c = A4(V))cr and Vtx, i.1 _< i
#£(~) ~ e # £i(q,)cr ,
¯ c E ~, where ¯ denotes a set of 0-forms, when there
is a O-formVJ in the set q, such that c E 0.
Wesay that two 0-forms axe equivalent, written
01 = ~02 if their corresponding ~p-sets axe equal, i.e.
OI = ~)2 if andonly if *(~’I) = $(~b2).
In a well-formed 0-form there are no exceptions
such that clauses it denotes are a subset of the
clauses denoted by another exception, i.e., ~b([Ei (~)]) ~Z
~b([£j(~b)]) for i,j where 0 _<i, j <#£(~b). It is
a simple matter to transform a ~b-form that is not
294 AIPS-2000
well-formed into an equivalent one that is well-formed.
Thus, from here on, we only consider well-formed ~forms. Note that. a well-formed #-form representation
of a ~b-set is not unique.
Prom now on we assume that every equation involving 0-forms or sets of ~b-forms actually denotes an
equality between the corresponding ~b-sets. I.e when
A and B are such expressions, A = B is a shorthand
for writing $(A) = $(B).
~b-form Logic
Wecan add ~#-forms to propositional logic by providing
an interpretation rule for them.
Aninterpretation, Z, is an assignment, of truth values
to all atoms of a theory. Interpretation rules extend
an interpretation by defining the truth value of every
sentence of a theory, not just atoms. Weadopt the
standard interpretation rules of the propositional logic
and add the following rule for interpreting a #-form or
a set of ~-forms, denoted below by [] .
Z([])
= A I(e),
(2)
cE$(n)
i.e. a single O-formor a set of #-form is interpreted as
a conjunction of all clauses in it.
Nowthat, we have defined an interpretation for #forms we can define entailment in a logical language
containing ~b-forms in the usual way. A model of a
logical formula is an interpretation that assigns true to
that formula. A formula a entails formula b if and
only if every model of a is also a model of b.
Given the definition of $, we can see that for any two
~-forms or sets of #-forms D, and 02, (~(r-I1)
= ~(r-Ii)
iff ( [~ ~ 02 and O2 V O~ ).
PSIPLAN Formalism
Worlds
and SOKs
A world state is a set of all literals that are true in the
world. Wedefine a domain proposition or simply a
proposition as either an atom or a ~b-form. Agent’s
state of knowledge, or SOKis a consistent set of domain propositions.
Representing
Actions
Actions axe ground and are represented in a fashion
similar to STRIPS. Eac~ action a has a name, .~(a),
a set of preconditions, 7~(a), and a set of domainliterals called the assert list, .4(a). Action preconditions
identify the domain propositions necessary for executing the action. The propositions in 7)(a) can include
3literals and quantified ~p-forms.
aRuling out other forms of non-quantified disjunction is
not a limitation, since any action schemathat has a non-
From: AIPS 2000 Proceedings. Copyright © 2000, AAAI (www.aaai.org). All rights reserved.
The assert list, also called the effects of the domain
action, identifies all and only the domainpropositions
whose value may change as a result of the action. We
assume that an action is deterministic and can change
the truth only finite numberof atoms, and thus any ~bform in the assert list defines a single negated literal.
State Update Procedure.
Actions cause transitions between the world states.
The agent’s SOKsmust evolve in parallel with the
world, and must adequately reflect the changes in the
world that occur due to an action. Correctness of an
S0K update guarantees that the SOKis always consistent with the world model, given a consistent initial
SOK. The other desirable property of the SOKupdate
is completeness: we would like the agent to take advantage of all information that becomes available and
not to discard what was previously known and has not
changed. Clearly, the correctness and completeness
properties of the SOKupdate, as well as soundness and
completeness of entailment within the state language
are prerequisites for a sound and complete planning
algorithm.
Weuse symbols w, w’, wl ... wn to refer to states of
the world, ~r W’ to refer to the sets of world states,
and s, s’, Sl ... 84 to refer to the agent’s SOK.
The correctness and completeness criteria axe best
formulated in the context of possible worlds.
We
define a set of possible worlds given a SOKs as the
set B(s) = {wlw ~ s}. Let do(a, W) denote the set of
world states obtained from performing action a in any
of the world states in W, and update(s, a) denote the
SOKthat results if the agent, performs action a from
SOKs. Wesay that the update procedure is correct
if[
B(update(s, a)) C do(a, B(s))
(3)
i.e. every possible world after performing the action a
has to have a possible predecessor.
The update procedure is complete if[
do(a, B(s)) C_ B(update(s,
(4)
i.e. every world obtained from a previously possible
world is accessible from the new SOK. This implies
that all changes to the world must be reflected in the
new SOK.
To achieve correctness of SOKupdates, the agent
must remove from the SOKall propositions
whose
truth value might have changed as the result of the
performed action.
quantified disjunction as its precondition, can be equivalently split into several actions, each of whichis identical
to the initial schema,but has only one of the disjuncts for
the preconditions.
In order to be complete, the agent must also add to
the SOKall facts that become known. The complexity of the SOKupdate thus depends critically on the
process of identifying the propositions that must be retracted to preserve correctness. In our language this
computation is reduced to computing the entailment,
which has polynomial complexity.
Before defining the world state transition caused by
actions, we introduce the operations of e-difference and
image . These operation come handy in planning, as
we will showlater.
For any two sets of ground propositions A and B we
define e-difference
and image (the image of A in
B), respectively, as follows:
B-A = {blbEB
^ Al~b}
A~B = {blbEB
^ At=b}
B-A is the subset of B that/8 not entailed by A while
A ~ B is the subset of B that is entailed by A. Thus,
(B "-A) and (A ~ always par tition B. Moreover, we
have the following equivalences.
1. B-A = B-(A~B)
and
2.
A~B = B-(B-A)
Determining image and e-difference between sets of
atoms is straightforward, and so we will not discuss
it. Between~b-forms, however, it is more complicated.
Wedefer discussing it until the Calculus Section..
Our agent can only execute an action a if its SOK
about the current state is s and s ~ 7~(a). To obtain
the agent’s SOKs after performing an action a, we first
remove all propositions implied by the negation of the
assert list, as only those propositions of s might change
their values after a. Wealso remove from the SOKall
redundant propositions, i.e. those that follow from the
effects of the action, and then add these effects to the
new state. The agent’s SOKafter executing action a
in the SOKs is described by the following formula.
~,pdateCs,a) = ((s- A-Ca))-A(a))(5
where for a set of propositions P, P- denotes the
set of the negations of propositions in P.
Theorem 1 The state of knowledge update procedure
(5) is correct and complete.
Wedo not present proofs in this paper due to space
limitations.
Example 2 For an example, we characterize
the
action a = mv(fig,/img,/tez),
which moves the
file fig from directory
/img into ~rex.
We
use T(z, PS) to represent that file z has type
postscript. Let 7~(a) = {In(fig,/img)}
which states
that fig must be in /img. Also, let A(a)
[-~In(fig,/irag),
In(fig,/tez)].
Webegin with an
SOK:
Babaian 295
From: AIPS 2000 Proceedings. Copyright © 2000, AAAI (www.aaai.org). All rights reserved.
s =
Split Ooal(¢~, ~bc ): Performwhen
"- q)e, ~’c are~b-forms
and
In(fig,/img), In(a.bmp, /img), T(a.ps,
- ~Penearlyentails ~bc- i.e.,
[-~In(x, d) V --T(x, PS) [ -~(x = a.ps) A -~(d =/img)],
[~(~)] ~ [~(W~)] but ~,. ~
lima) I = fig) ^ = a.bmv)]
a = mv(fig, fling, flex) is executable in s, and the
resulting SOKis:
In(fig, itexj, [n(a.bmp, lirng), T(a.ps, PS),
[-,In(z,/img) i-~(x = a.bmp)],
[-~In(x, d) V -~T(z, PS) [ ~(x = a.ps)
-~(d = fling) ^ -~(x = fig, d = flex)]
Note that s contained -~In(fig, /tex) V -~T(fig,
and that we added In(fig,
flex) when determining
s’. If our update rule retained -~In(fig,/tex)V
-~T(fig, PS) in s’, then in s’ we could perform resolution and conclude that -~T(fig, PS). However.
this would be wrong because we have no information on fig being a postscript
file or not. Instead, our update rule deletes any clause that is entailed by -~ln(fig,/tex),
and so s~ does not contain
-~In(fig, flex) V -~T(fig, PS).
s’ =
The Planning
Problem.
As usual in a planning problem we are given a set of
initial conditions 27, a set of goals ~ and a set of a~-ailable actions. 27 and ~ are sets of domainpropositions.
A solution plan is a sequence of actions, that is executable and transforms an)" world state satisfying the
initial conditions into a world state satisfying the goal.
Our planner operates in SOKsinstead of worlds.
The correctness and completeness of the update function, however, guarantees that it returns all and only
solution plans, provided the initial SOKso is equal to
27 plus all propositions obtained from 27 by performing
resolution. A planner is called sound if and only if
it returns only solution plans, and complete, if it returns every solution plan. Presented in the next section
PSIPOPis a sound and complete planning algorithm.
PSIPOP Algorithm
Weassume the reader is already familiar with SNLPstyle planning (McAllester & Rosenblitt 1991).
made a few changes to the standard algorithm so
that it easily generalizes to handling ¢-forms. These
changes arise when ~-forms are added to the state and
action description language. Since a link bet~en two
¢-forms actually represents a multitude of links between ground clauses of the source and target ¢-forms,
we need to introduce new techniques of establishing
and protecting such links. These techniques are based
on the theory of 0-form entailment, presented in the
next section.
There are three important changes to the POP.
296 AIPS-2000
1. Partition ~b~into ~b~= ¢~ ~, ~’¢ and (¢¢-t%).
’1 S~. to the links.
2. AddS, ¢~2~
3. Foreach ¢ E (~’c--t/’e),
add < W, Sc > to the open goals.
Split Link(~be, ~P~ ):Perform when
- effect A on St threatens S, or.~" S~- i.e.
([-,A]~ ~,)~ ~.,
1. AddS~-~ St. and St -4 S~. to orderings.
2. Partition ~b, into [-~A]~, ~/., andtg~ = ~,-" [--A].
3. Partition¢c into (I--A]I> ~be)t> ~bcand
¢~= ¢~’-([-A]~,).
4. Remove
original link S~ u’~"" S~ from links.
’-~ S~to links.
5. AddS, o~L~
6. Foreach~bE ((I--A]t> ~/~e)~, ~t’e),
add < ~b, Sc > to the open goals.
Figure 1: Split Goal and Split Link Procedures
Change 1. Causal links now have both source and
target conditions, which may differ. The source condition must entail the target condition. For example,
we may have step S~ with effect 9~ and step $2 with
precondition 92 where~Pl states that are no. files in directory/psdir except Postscript files. 92 requires that
there are no files of type TEXin/psdir owned by Joe
except a.ps.
V -T(x,
¢2 = [-~In(x,/psdir)
V-~T(x, T EX) V -~O(x,
- Cx=a.tex)]
y)
-~(y =
1~ I = I-In(z,/psdir)
PS)]
,/oe)
Clearly, if there are no TEXfiles in/psdir, then there
are no TEXfiles owned by Joe in it, so 9~ ~ ¢2 and
we can have a causal link from ~Pl on S~ to ¢2 on 82.
Thus, causal links between ~b-forms actually represent a set of causal links between each clause of the
goal and the supporting clause in the source.
Change 2. Whenwe cannot find a single ~p-form that
entails all clauses in a goal 9-form, we, however, must
find a 9-form that entails most of them. Consider the
goal of not having files of any format owned by Joe in
/psdir, represented by 9z
92 = [-.In(x,/psdir) V -.T(x, y) V -.O(x,
Assumeagain that we have 91 as an effect on step St
and 92 as a precondition on step $2 where
9~ = [-,In(z,/psdir)
V -~T(x, Y) I-’(Y =
Clearly, from ¢~ we can conclude that none of Joe’s
files are in/psidir except for his Postscript files, about
which 9~ states nothing.
From: AIPS 2000 Proceedings. Copyright © 2000, AAAI (www.aaai.org). All rights reserved.
In cases like this one where ~bx ~: ~b2 but where ~bx
nearly entails ~b2, we try splitting the goal ~b2. ~1
nearly entails ~]~ iff [A4(~bl)] ~ [A4(~b2)]. In
cases, we split ~b2 into two parts:
¯ The precise portion of ~b2 that is entailed by ~bx--this
is the imageof ~bx in ~b2, Ox~ ~b2---and
¯ The remainder of ~b2--this is precisely ~b2-~bl.
Once split, we add a causal link from ~bl on Sx to
(~bl ~b2) on $2, and weare left with (~b2-" ~bl) on $2
still needs to be linked. Fortunately, calculating both
image and e-difference is straightforward and results in
a finite set of ~b-forms,each of whichare strictly smaller
than the original ~b-form goal in a well founded way.
Applying Goal Splitting (Fig. 1) to our example,
split ~b2into:
¯ ~
=
~1
~
~2
=
[-In(z,/psdir) V -,T(z, y) V -~O(z, I-’(Y = PS)]
which are all the files in/psdir ownedby Joe except
for Postscript files, and
¯ ~]
=
~2-" ~
=
[--In(x, [psdir) V --T(x, PS) V -,O(x, Joe)], which
are all the Postscript files in/psdir ownedby Joe.
Next, we add a causal link $1 ~ $2 and we are
left with ~b] unsupported. Thus, much of ~b2 is now
supported except for ~b].
Change 3. The final change to POP adds a new way
to resolve threats. A threat is any effect of an action,
that results in the removal of the source condition in
the SOKduring the update (see (5)). In our formalism,
only a ground atom can threaten a link between ~bforms, and conversely, only a ~b-form can threaten a
link between ground atoms. In the former case, we
add a new threat resolution methodcalled link splitting
(Fig. 1).
For an atom to threaten a link, it must remove from
the source ~b-form proposition(s) that support some
proposition(s) in the target ~b-form. More formally,
atom A maypose a threat to the link from ~bl to ~2 if
([-a] ~ ~1)~ ~2¢.
Let there be a link from effect ~bl on step S1 to precondition ~b2 on step $2. Moreover, let atom A be an
effect of step S where A threatens the link. Wewill
refer to a ~b-form constructed out of the negation of A,
namely, ~bA= [-~A], whichis a singleton set.
The threat resolution method does the following.
¯ ~bl on step $1 is partitioned into ~b~ = ~bl-~bA and
~ = ~bA~ ~bx. Note that ~b~ is precisely the subset
of ~b~that is not threatened by A and that ~b2 is the
residual of ~.
¯ ~2 is replaced by ~b~ = ~bl ~ ~ and ~ = ~b~-~b~.
Note that ~b~ is precisely the subset of ~b2 that is
now supported by ~b] and that ~a2 is precisely the
residual of ~.
¯ The original link from ~bi to ~b2 is replaced by a link
from ~bl to ~b~. Condition A on S no longer is a
threat.
¯ Newsupport must be found for ~ on step $2.
Calculus
In this section we sketch howto determine entailment,
image and e-difference. These calculations are somewhat complex and we do not have space to present
them fully. A complete description can be found in
(Babaian & Schmolze 1999). For the reader who
not interested in these methods, this section can be
skipped.
Everywhere below we make a su~iciently large domain assumption, i.e. that the object domain contains
more objects that are mentioned in all of the participating ~b-forms. Certainly, a domainthat is only partially knownis sufficiently large.
Determining
Entailment
Domain ~b-forms. The critical factor in keeping ~form reasoning tractable is given by the following Theorem, that states, essentially, that we do not have to
examine combinations of ~b-forms when checking ~bform entailment.
Theorem2 Let ~L’I,...,
~bn be simple ~b-forms and
let ~b be an arbitrary ~b-form. {~bl,..., ~bn} ~ ~b iff
3i. (1 < i < n) A (~bi ~ [M(~b)]).
Thus, if we ignore exceptions, then to showthat a set.
of ~b-forms entails ~, we need to find only one ~b-form
in the set that entails [fl4(~b)].
Checking if ~bi ~ [fl4(~b)] is, as it turns out from
the next Theorem, is just a matter of finding a subsetmatch between the the main parts, as both ~b-forms are
simple. At the heart of this result is the observation
that given two ground clauses, C~ and C2, C~ ~ C2 iff
C~ c_ C~. Here, we are treating each ground clause as
a set of literals.
Theorem3 Given two simple ~b-forms, ~b~ and ~b~,
~bl ~ ~b2 iff there exists a unifier, ~ such that
~(~)~ _c ~(~2).
Note that there can be more than one way a clause
can subset-match onto another clause. For example,
matching -,P(x) onto ",P(a) V’,P(b) V --Q(y) produces
two different substitutions: (x = a) and (x =
The next Theorempresents necessary and sufficient
conditions for entailment between ~b-forms.
Theorem 4 Let ~p~,... ,~bn and ~b be arbitrary ~bforms. {~b~,... ,~bn} ~ ~b iff there exists a k,1 < k < n,
such that:
¯ [A4(~)] ~ [fl4(~b)] (i.e.,
[A4(~b)]),
the main part of ~ entails
Babaian
297
From: AIPS 2000 Proceedings. Copyright © 2000, AAAI (www.aaai.org). All rights reserved.
¯
The last requirement in Theorem 4 requires some
cxplanation. While [/I/L(¢k)] ~ [.~(¢)] , the
ceptions of ~k weaken Ck. Thus, each clause in
~P’-¢k must be entailed by some other C-form in
{~I,’’’,~k--1,~k-l-l,’’’,~n}"
We discussmethodsof computing
the e-difference
in a latersection.
Herewe wouldjustliketo notice
thatentailment
in PSIPLAN
is easilydecided
andhas
a timecomplexity
that.is polynomial
in thenumber
of
propositions,
maximumnumberof exceptions
and the
maximumnumberof variables
usedin a C-form.
Note also the following simple facts.
Given two ground atoms A and B, A entails B, written A ~ B, iff A = B.
Given an atom A and C-form ¢, A can never entail
~) and ¢ can never entail A.
Determining
Image and E-Difference
Among C-forms
The image and difference operators between C-forms,
in general, are complex. Important for the algorithms
presented in this paper are the facts that both operations produce sets of C-forms, the time complexity of
C-form image and e-difference is polynomial the maximum number of variables used in a C-form and the
maximumnumber of exceptions. Here we only state
the key results to provide the reader with some intuition about C-form calculus. The full account of this
calculus can be found in (Babaian & Schmolze 1999).
Wedefine MGU(A,B, V) as the most general unifier
of A and B using only the variables in V. I.e., For a =
MGU(A,B, V), Ao = Be and is a most gen eral suc h
unifier. In a similar fashion, we define MGUc_
(A, B, V)
as the set of all a such that Ao C_ Ba and o is a most
general such unifier.
MGU(A,B) and MGUcA, B)
are defined similarly, except they do not restrict the
bindings to any" particular set of variables.
Theorem 5 For any" Ct, ¢2 - simple C-forms ¢i ~¢2 =
Io¯
So computing
the imageof one simple~b-formin another simple C-formamountsto finding either a unifier,
or a set of subset unifiers of their main parts and appropriately instantiating the main part of the second
C-form.
As we will see shortly, to compute e-difference we
first perform the set or subset unification procedure
and then add the resulting substitution(s) to the set
exceptions of the second C-form. Before adding those
substitutions,
we preprocess them to conform to the
syntax of C-forms, i.e. we removeall bindings on variables in Y(¢1), add a -1 sign in front of each o. That
is the purpose of the E’ in the next Theorem.
298 AIPS-2000
Theorem6 For any Ct, ¢2 -C-forms such that ¢1 and
is simple
if ¢i ¢2,
v z’]}
otherwise.
¢2-’¢i= { ¢’
where 3~r = {-’,a~la
¯ MGU¢(./VI(¢t),.Ad(¢2))
Note, that the first case is actually subsumedby the
second, because if ¢1 ~ ~2, MGUc_(./~I(¢I),.M(¢2))
contains the substitution
a which subset matches
fl4(¢t) onto fl4(¢2), i.e. a only uses variabIes
V(¢1). In such case the added exception ~ =-- 0 de notes the whole [~1(¢2)], thus leaving the resulting
C-form equal to 0.
Also, when MGUc(.A4(¢I),,~I(¢2)) is empty, E~ is
empty and ¢2-" Ct = ¢2.
The image and e-difference computations in the general case are reduced to computing the operations on
the simple C-forms.
Related
Work
Wehave already briefly discussed the work of (Golden,
Etzioni, & Weld 1994; Etzioni, Golden, & Weld 1997;
Golden 1998) in the introduction. PUCCINI?? has
richer action and goal languages, handles sensing actions and execution. Hut due to the incompleteness of
the LCWreasoning, it is incomplete even when it does
no sensing.
Conformant Graphplan (Smith & Weld 1998) and
SGP (Weld & Anderson 1998) are propositional open
world planners that consider every possible world and
thus rely" on the domainof objects being sufficiently
small.
In another related work, Levy (Levy 1996) presents
a method for answer-completeness, which determines
whether the result of a database query is correct when
the underlying database is incomplete. For this work,
the database is relational and incompleteness means
that it may be missing tuples. The incompleteness
is expressed with LC constraints, which are based on
LCWsbut which are considerably more expressive. In
fact, Levy’s LCs can easily represent the information
in LCWswith exceptions, which is roughly the expressive power of C-forms. (Friedman & Weld 1997) expands on (Levy 1996) with a richer set of LC constraints and an algorithm that determines whether a
given information-gathering plan subsumes another.
Both of these works address the answering of queries
in an unchanging database. They do not,however,
address a changing world, much less the planning of
an agent to change it.Also, PSIPLANneeds several
operations--e.g., entailment, e-difference and image.....
From: AIPS 2000 Proceedings. Copyright © 2000, AAAI (www.aaai.org). All rights reserved.
for planning that do not arise when only answering
queries.
Conclusions and Future Work
Wehave presented PSIPOP, a sound and complete partial order planning algorithm that does not make the
closed world assumption and that can represent partially closed worlds. The key idea is the use of ~forms to represent quantified negative information and
to integrate ~-forms into POPsuch that it adds only
polynomial cost algorithms. Wethus argue informally
that, even with our expanded representation, we can
keep the complexity of planning within NP if we have
a finite language and if we bound the length of plans.
We have developed an extended formalism and planning system PSIPLAN-S,that uses an extension of the
PSIPOPalgorithm to handle information goals, sensing actions, information loss, conditional effects and
execution. The system has been implemented in Common Lisp and has performed successfully on numerous
examples. The extension of the presented algorithm to
a lifted version with conditional planning and execution is straightforward.
Future work is already in progress. Wewill continue to explore richer representations for ~-forms. We
will also incorporate into the formalism actions that
both sense and change the world. Weare investigating
methods for efficient integration sensing, conditional
planning and plan execution. Wewill also examine
application
of PSIPLANto SAT planning (Kautz
Selman 1996).
Acknowledgments
The authors thank Keith Golden and Neal Lesh for
their commentsthat helped improve this paper.
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